Diameter of Random Cayley Graph of Z_q

a r X i v :m a t h /0609620v 1 [m a t h .P R ] 21 S e p 2006Diameter of Random Cayley Graph of Z qGideon Amir∗Ori Gurel-Gurevich†February 2,2008AbstractConsider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators.For k fixed,we prove that thediameter of said graph is asymptotically (in q )of order k√∗Weizmann Institute,Rehovot,76100,Israel.gideon.amir@weizmann.ac.il †Weizmann Institute,Rehovot,76100,Israel.ori.gurel-gurevich@weizmann.ac.il1Various aspects,most notably the mixing time,of random random walks on differentfinite groups have been studied.(See[1],[5]for some examples, and[2]which gives a comprehensive survey).The results usually refer to Abelian groups in varying degrees of generality,from the cyclic group Z n up to generalfinite Abelian groups.Roichman([6])notes that the diameter of the random Cayley graph is bounded by a constant times the mixing time,and applies this bound to the case of general groups of order n and k=⌊log a n⌋.The resulting bound is aq⌉}in Z q(where q is prime).The diameter of this Cayley graph is2√q⌋}.Thediameter of this Cayley graph is now q.Put another way,the diameter does not change when adding or removing0from the generating set but the mixing time might change considerably.Another point of notice is the question of symmetric vs.asymmetric generating sets.The results in[3]are for asymmetric generating sets,i.e.S contains just the k randomly chosen generators.We might as well ask about the mixing times and diameters w.r.t.equally to the symmetric case.The proofs below suggest that the behavior of the diameter in the symmetric and asymmetric cases should be the same up to a multiplicative constant of about2k.It should be noted that the asymptotic behavior of both the mixing time and the diameter,both in the symmetric and asymmetric case are the same as in the case of a k dimensional tori of volume q.This is not coincidental, the structure of the Cayley graph of G w.r.t.S is actually that of Z k,modulo some k-dimensional lattice,which contains the lattice of all multiples of q. Perhaps the mixing time results of[3]could be proved in a more elementary manner using that perspective.2The Main ResultsLet G=Z q the cyclic group of order q,a prime number.Let g1,..,g k be random generators chosen uniformly and independently from G.Denote by Diam(q,k)the random variable which is the diameter of the resulting Cayley graph.All asymptotic results are w.r.t.q while k remainsfixed. Obviously,the diameter must be at leastΩ(k√q)in the following sense:Theorem1For every k and everyǫ>0there exists C such that for all prime qP rob(Diam(q,k)>C k√q)>ǫ33Proof of Theorem1For any x∈Z q andias the event i1g1+i2g2+..+i k g k=x(mod q).Let A x= i and Let A= x∈Z q A x.By standard abuse of notation,we will not distinguish between an event and its indicator variable.We ask for which L is P(A)→1?Obviously,for any0,E(A xiA xi andiA xi and i=λi ,A xiA xi)E(A x i∈L kA xi∈L kE(A xq.V ar(A x i)=1q)<1i∈L k V ar(A xi=i,A xq.4Using Chebyshev’s inequalityP (A x=0)=P (B x=0)≤P (|B x−E (B x)|≥E (B x))≤V ar (B x )(L k /q )2=qL kLet B =x ∈Z q A x ,i.e.the number of points covered by a ball of radius L .Fix C >2and let L =k √Lk =1−1C).Since B is never more than q ,P (B >qC.If a ball of radius L covers more than half the graph,then the graph has at most diameter 2L .This is because a Cayley graph is vertex transitive so all balls of radius L have the same volume and therefore any two such balls intersect.Thus we conclude,that given ǫ>0the probability that the diameter ofthe Cayley graph is more than 2kǫq is less than ǫ.4Proof of Theorem 2Fix D <1.Let L =k√iand A 0as in the previous section.As before,ifj are linearly independent over Z q then A 0j areindependent events.Ifj are linearly dependent then A 0jare infact identical.How many distinct events do we have among {A 0i andi =cq we geti 0j 1=i 1j 0.Take allTherefore,among the vectors taken above every two are linearly independent, so the corresponding events are pairwise independent.Given L how many pairs i0,i1<L are coprime?It is a well known result that P rob(i0⊥i1)tends to,and is always greater than6/π2which is biggerthan1/2(see[?]).i∈IA0i∈IE(A0qV ar(A0q (1−1212V ar(X)=Dqq(1−12(1−12(1−12)Since X is nonnegative,from Cauchy-Schwartz inequality we get E(X)2=E(X1X>0)2≤E(X2)E((1X>0)2)=E(X2)P rob(X>0)ThereforeP rob(X>0)≥E(X)242(1−12)≥DFrom lemma4we get that the probability of A0is bounded away from 0regardless of q.Next we shall show that if A0occurs then many differentLemma 5Let C be such that k k C k −1D <1.If A 0occurs thenDiam (q,k )>ki ∈{1..k√ioccurs.Ifj ′differ in amultiple ofj ∈{1..k√j ′such thatj 1g 1+j 2g 2+..+j k g k =j ′1g 1+j ′2g 2+..+j ′k g k (modq )and j ′t ≤k √j ′is less than k (k √Dq ).If this number is less than q then the diameter cannot be k√To warp up the proof,given C take any D <k −k C 1−k and let ǫ=D/3.From lemma 4we conclude that A 0(w.r.t D )occurs with probability at leastǫ,in which case,by lemma 5Diam (q,k )>k√k√would consist of k independent random variables,g1,..,g k,distributed uni-formly in[0,1].Let A d be the set of all points i1g1+..+i k g k(mod1)with i1,..,i k integers less then some d.Let a d be the largest gap in A d,i.e.the)is not[0,1].The asymptotic behavior of a d largestǫfor which A d+(−ǫ2(or rather,its inverse)should be closely related to the asymptotic behavior of Diam(q,k).AcknowledgementsWe would like to thank our advisor,Itai Benjamini,for suggesting this ques-tion and for useful discussions.References[1]N.Alon,Y.Roichman(1994)Random Cayley graphs and expanders.Random Structures and Algorithms,5(2),271-284[2]M.V.Hildebrand(2005)A survey of random random walks onfinitegroups.Probability Surveys2,33-63[3]M.V.Hildebrand(1994)Random walks supported on random points ofZ/n Z.Probability Theory and Related Fields100,191-203[4]D.Wells(1986)The Penguin Dictionary of Curious and InterestingNumbers.Middlesex,England:Penguin Books,28-29[5]D.B.Wilson,Random random walks on Z d2.Probability Theory and Re-lated Fields108(1997),no.4[6]Y.Roichman,On random random walks.Ann.Probab.24(1996),no.2,1001–1011.8。